Discrete Math Validating Argument Using Deduction Method

Question

I am lost trying prove that the expression below is a valid argument using the deduction method (that is using equivalences and rules of inference in a proof sequence).

(∃x)[A(x)∧B(x)]→(∃x)A(x)∧(∃x)B(x)

Answer 1

See List of rules of inference or :

• Kenneth Rosen, Discrete mathematics and its applications (7th ed), pages 72 and 76 for the rules :

1) $(∃x)[A(x)∧B(x)]$ --- premise

2) $A(y)∧B(y)$ --- from 1) by Existential Instantiation : $y$ not occurring into 1)

3) $A(y)$ --- from 2) by Simplification

4) $B(y)$ --- from 2) by Simplification (or Conjunction Elimination)

5) $(∃x)A(x)$ --- from 3) by Existential Generalization

6) $(∃x)B(x)$ --- from 4) by Existential Generalization

7) $(∃x)A(x)∧(∃x)B(x)$ --- from 5) and 6) by Adjunction (or Conjunction introduction

8) $(∃x)[A(x)∧B(x)] \to (∃x)A(x)∧(∃x)B(x)$ --- from 1) and 8) by Conditional Introduction